Conditional Probability — Practice

Practice bank for UCAT Decision Making questions involving conditional probability.

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Quick-Reference: Conditional Probability

Definition

$P(A|B) = \frac{P(A \cap B)}{P(B)}$P(A∣B)=P(B)P(A∩B)​

$P(A|B)$P(A∣B) is the probability of A given that B has occurred.

Key Distinction

$P(A|B) \neq P(B|A)$P(A∣B)=P(B∣A) in general.

• P(positive test | disease) = sensitivity
• P(disease | positive test) = PPV
• These are very different numbers!

From a Table

If given a two-way table, conditional probability is:

$P(A|B) = \frac{\text{count in both A and B}}{\text{total count in B}}$P(A∣B)=total count in Bcount in both A and B​

Independence

Events A and B are independent if $P(A|B) = P(A)$P(A∣B)=P(A). That is, knowing B occurred does not change the probability of A.

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The Two-Way Table Method

Given a scenario with two characteristics, build a 2×2 table:

	Characteristic B+	Characteristic B−	Total
Characteristic A+	Both	A only	Row total
Characteristic A−	B only	Neither	Row total
Total	Col total	Col total	Grand total

Then:

• $P(A|B) = \frac{\text{Both}}{\text{Column total for B+}}$P(A∣B)=Column total for B+Both​

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Common Pitfalls